How did Coulomb determine the law of electric force between two point charges?

(N/A) Coulomb assumed the charge on a metallic sphere is $q$. If the sphere is placed in contact with an identical uncharged sphere,the charge will be distributed equally over the two spheres. By symmetry,the charge on each sphere will be $\frac{q}{2}$.
By repeating this process,we can obtain charges of $\frac{q}{2}, \frac{q}{4}, \frac{q}{8}, \dots$ on the spheres.
Coulomb varied the distance $r$ for a fixed pair of charges and measured the force $F$ for different separations. He observed the relation:
$F \propto \frac{1}{r^{2}} \quad (1)$
He then varied the charges $q_{1}$ and $q_{2}$ while keeping the distance fixed. By comparing the forces for different pairs of charges,he established the relation:
$F \propto q_{1} q_{2} \quad (2)$
Combining these,the electric force between two point charges is given by:
$F \propto \frac{q_{1} q_{2}}{r^{2}}$
Therefore,$F = k \frac{q_{1} q_{2}}{r^{2}}$,where $k$ is the Coulomb constant.